The magic word is DISPLAY. In the X window system, a display consists
(simplified) of a keyboard, a mouse and a screen. A display is managed
by a server program, known as an X server. The server serves displaying
capabilities to other programs that connect to it.
A display is indicated with a name, for instance:
DISPLAY=light.uni.verse:0
DISPLAY=localhost:4
DISPLAY=:0
The display consists of a hostname (such as light.uni.verse and
localhost), a colon (:), and a sequence number (such as 0
and 4). The hostname of the display is the name of the computer
where the X server runs. An omitted hostname means the local host. The
sequence number is usually 0 -- it can be varied if there are multiple
displays connected to one computer.
If you ever come across a display indication with an extra .n
attached to it, that's the screen number. A display can actually have
multiple screens. Usually there's only one screen though, with number
n=0, so that's the default.
Other forms of DISPLAY exist, but the above will do for our purposes.
For the technically curious:
hostname:D.S means screen S on display D of host
hostname; the X server for this display is listening at TCP
port 6000+D.
host/unix:D.S means screen S on display D
of host host; the X server for this display is listening
at UNIX domain socket /tmp/.X11-unix/XD (so it's only
reachable from host).
:D.S is equivalent to host/unix:D.S, where host
is the local hostname.